u 3 x, t
ð Þ ffi w x 1 , t
ð
Þ,
u 1 x, t
ð Þ ffi x 3 ψ x 1 , t
ð
Þ,
φ x, t
ð Þ ffi φ x 1 , t
ð
Þ,
ð6:103Þ
where w(x 1 ,t) is the bending displacement (deflection) and ψ(x 1 ,t) the shear deformation accompanying bending. The relevant strain and electric field components are
S 1 ¼ S 11 ¼ u 1,1 ¼ x 3 ψ ,1 ,
S 5 ¼ 2S 31 ¼ u 3,1 þ u 1,3 ¼ w ,1 þ ψ,
E 1 ¼ Àφ ,1 :
ð6:104Þ
For bending in the x 1 -x 3 plane, the main stress components are the normal stress T 1
and shear stress T 13 ¼ T 5 . We introduce the following stress relaxation for thin
beams:
T 2 ¼ T 3 ¼ T 4 ¼ T 6 ffi 0:
ð6:105Þ
For the piezoelectric middle layer, the constitutive relations for the relevant strain
and electric displacement components are
S 1 ¼ s
1
ð Þ
11 T 1 ,
S 5 ¼ s
1
ð Þ
55 T 5 þ d 15 E 1 ,
D 1 ¼ d 15 T 5 þ ε
1
ð Þ
11 E 1 :
ð6:106Þ
We invert Eq. (6.106) 1,2 for expressions of stresses in terms of strains and substitute
the resulting expression for T 5 into Eq. (6.106) 3 . This gives
T 1 ¼ T 11 ¼ c
1
ð Þ
11 S 1 ¼ c
1
ð Þ
11 x 3 ψ ,1 ,
T 5 ¼ T 31 ¼ c
1
ð Þ
55 S 5 À e 15 E 1 ¼ c
1
ð Þ
55 w ,1 þ ψ
ð
Þþe 15 φ ,1 ,
D 1 ¼ e 15 S 5 þ ε
1
ð Þ
11 E 1 ¼ e 15 w ,1 þ ψ
ð
ÞÀε
1
ð Þ
11 φ ,1 ,
ð6:107Þ
where Eq. (6.104) has been used. The one-dimensional effective material constants
for thin beams in Eq. (6.107) are
c
1
ð Þ
11 ¼ 1=s
1
ð Þ
11 , c
1
ð Þ
55 ¼ 1=s
1
ð Þ
55 , e 15 ¼ d 15 =s
1
ð Þ
55 ,
ε
1
ð Þ
11 ¼ ε
1
ð Þ
11 À d
2
15 =s
1
ð Þ
55 :
ð6:108Þ
Similarly, for the semiconductor layers, the one-dimensional constitutive relations
for thin beams are
6.4 Bending of Beams with e 15
163